COMBINATION. The combination of quantities is the showing how often a lessnumber of things may be taken out of a greater, and combined together, without considering the order they stand in thus-out of the letters abc there are three combinations of two, that is, ab, ac, and bc. RULE. : 1. Take the series 1, 2, 3, 4, &c. up to the number to be taken at a time, and find the product of all the terms. 2. Take a series of as many terms, decreasing by 1, from the givon number out of which the combination is to be made and find the product of all the terms. 3. Divide the last product by the first, and the quotient will be the answer required. EXAMPLES. 9 12 11 132 10 6 1320 4 24 11880 5 8 120 95040 2. How many combinationsTM can be made of 6 letters out of the 24 letters of the alphabet? Ans. 134596. 3. A butcher bargained with a farmer (well skilled in numbers) for a dozen sheep, (at 2dolls. per head) which were to be picked out of 2 dozen; but, being long in choosing them, the farmer told him that if he would give him a a cent for every different dozen which might be chosen out of the 2 dozen, the should have the whole, to which the butcher readily agreed; what did they cost him? Ans. $27041.56cts.. 4.. A general was asked by his 5040 )3991680(792 Ans. king what reward he should con-fer on him for his services; the general required a penny for every file of 10 men in a file, which he could make out of a company of 90 men; what sum did it amount to? Ans. 23836022841€ 5s. 3d. OF MENSURATION. Mensuration teaches us how to find the area or superficial content of any plain surface or superficies. Also, the solidity or cubical content of any solid body. N. B.-1. A superficies or surface is an extension of two dimensions, namely, length and breadth, but is not considered as hav ing thickness. 2. A solid body of any thing has three dimensions, viz., length, breadth, and thickness. TABLES OF DIFFERENT MEASURES. I. OF LINEAL MEASURES. make 1 foot. make 1 yard. II. OF SQUARE MEASURES. 12 inches 3 feet 6 feet make 1 fathom. make 1 pole or 311 yards rod. 40 poles 8 furlongs make Ì furlong. 1600 poles make 1 furlong. 64 furlongs make 1 mile. N. B. The chain made use of in measuring land, commonly called Gunter's chain, is 4 poles, or 22 yards in length, and contains 100 equal links, each link being of a yard, equal to .66 decimals of a foot, or 7 inches and .92 decimals of an inch long. An acre of land is equal to 10 square chains; that is, 10 chains in length and 1 in breadth; 4840 square yards, 160 square poles, or 100,000 square links (each being the same in quantity) make 1 acre. 1728 cubic inches 27 cubic feet 166 cubic yards III. OF CUBIC MEASURES. make 1 cubic foot. 128 cubic feet, or 8 ft. in length,make 1 cord of wood. 4 in breadth, and 4 in height 242 cubic feet, or 161 ft. long, 1 thick and 1 in height 231 cubic inches 282 cubic inches 2684 cubic inches make 1 perch of stone. make 1 gallon wine measure. make 1 gallon ale measure. make 1 gallon dry measure. MENSURATION OF SUPERFICIES. The area of any plain surface is estimated by the number of squares contained within the bounds thereof, without any regard to thickness. The side of a square may be an inch, a foot, a yard, link, a chain, or a pole, whichever may suit best in the opinion of the calculator; and hence the area is said to be so many square inches, square feet, square yards, square links, &c. according to the dimension of the measuring unit. PROBLEM 1. To find the area of a square.... RULE. Multiply the side of the given square into itself, and the product will be the area or superficial content, and of like name with the denomination of the measuring unit, let it be inches, feet, or yards respectively. EXAMPLES. 2. Required the area or superficial content of a square floor, each side of which is 16 feet. 16 feet long. 16 feet wide. 96 16 Ans. 256 square feet, because the measuring unit is one foot. 3. What is the area of a square field whose side is 37 chains and 25 links? Ans. 138a. 3ro. Ipo. 4. What is the area of a square field whose side is 250 yards? Ans. 12a. 3r. 26p.+560. 5. What is the area of a square floor that measures 24 feet 6 in. 9 sec. every way? Ans. 603 sq. feet, 3in. 9sec. 6 thirds and 9 fourths.-The student is requested to prove the answer to be right by Practice and Duodecimals. PROBLEM 2. To find the area or superficial content of a parallelogram or long Multiply the length by the breadth, and the product will be the area or superficial content required. EXAMPLES. 1. What is the superficial content of a plank that is 14 feet 6 inches long, and 4 feet 9 inches wide? Ans. 68ft. 10in. 6 sec. 2. What is the area of a field, in acres,, whose length is 14 chains 50 links, and breadth 9 chains 75 links? Ans. 14a. Oroo. 22per.. PROBLEM 3. When the breadth of a plank is given, to find how much in length will make a square foot. RULE. Divide 144 square inches by the breadth, and the quotient will be the answer required. Multiply the length by the perpendicular height, and the product will be the area required. NOTE. A rhombus has four equal sides, two of its angles are greater and two less than the angles of a square. Multiply one of the longest sides by the perpendicular height, and the product will be the area required. NOTE-A rhomboid is a figure whose opposite sides and angles; are equal. AN EXAMPLE. Suppose ABCD to represent a rhomboid, the two longest sides, namely, AB and CD, each being 10 chains 50 links, and the perpendicular height AE 7 chains 60 links; re- D quired, the area in acres, &c. acres. E ch. li ch. lê. 10 PROBLEM 6. To find the area of a triangle. RULE. 1. If it be a right angled triangle, multiply the base by half the perpendicular, or half the base by the whole perpendicular, and the product in either case will be the area or superficial content required. 2. If it be an oblique angled triangle, (whether obtuse or acute,) multiply the base by half the perpendicular, let fall from the opposite angle, or half the base by the whole perpendicular, and the product in either case will be the area required; or, multiply the whole base by the whole perpendicular, and half the product will be the area. NOTE.-An obtuse angle contains more than 90 degrees, and an acute angle less. EXAMPLES. 1. Suppose the base AB of the right angled triangle ABC to measure 10 feet 9 inches, and the perpendicular BC 7 feet 3 inches, what is the area? in. ft. in. 3=1109-base AB. 7 3in. per. BC. 75 3 2 83" 2)77 11 3 38 11 7 6 Ans. Prove theanswer to be right by decimals and duodecimals. 2. The oblique angled triangle. ABC being given, draw a perpendicular from the obtuse angle at C to the base AB, and that perpendicular is the height of the triangle. The base AB is supposed to be 84 poles and the perpendicular CD 56; what is the area in acres, &c.? T C D Ans. 14 acres 2 roods 32 poles |